As the extension from the SM for the strongly first-order EWPT, effective field theories have often been considered like the standard model effective field theory (SMEFT) [1]}, [2]}, [3]}, [4]}, [5]}, the Higgs effective field theory (HEFT) [6]} and its variations such as the nearly aligned Higgs effective field theory (naHEFT) [7]} etc.
In addition to the effective field theory approaches, studies in renormalizable models with an extended scalar sector have also been performed intensively [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}, [34]}, [35]}, [36]}, [37]}, [38]}.
In both approaches, it has turned out that there is a strong correlation between physics to satisfy the condition of strongly first-order EWPT and to make a large deviation from the SM prediction in the triple Higgs boson coupling (\(hhh\)  coupling) [1]}, [14]}.
For example, the deviation larger than about 20\(\%\)  is predicted by the condition of strongly first-order EWPT in the two Higgs doublet model [14]}, [42]}, [43]}.
Such a large deviation in the \(hhh\)  coupling is expected to be measured at future collider experiments, such as High-Luminosity LHC (HL-LHC) [44]} and the International Linear Collider (ILC) [45]}, [46]}, [47]}, by which a scenario of strongly first-order EWPT can be tested.
